Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by using Bayesian inference and estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory.

Source: WIKIPEDIA

• Original error estimate
• Error in estimate
• Calculate Kalman gain
• Calculate current estimate
• Previous estimate
• Error in data (measurement)
• Calculate new error in estimate
• Updates estimate

• Original estimate
• Original error in estimate

original.estimate <- 4
original.error    <- 2

current.estimate <- original.estimate
current.error    <- original.error

\(\text{kalman gain} = KG = \frac{E_{EST}}{E_{EST} + E_{MEA}}\)

kalman.gain = current.error / ( current.error + measured.error )

\(\text{measurement} = MEA\)

previous.estimate <- current.estimate

• \(\text{current estimate} = EST_t\)
• \(\text{previous estimate} = EST_{t-1}\)

\[EST_t = EST_{t-1} + KG [MEA - EST_{t-1} ]\]

current.estimate = previous.estimate + kalman.gain * (measurement - previous.estimate)

\[E_{EST_t} = \frac{E_{MEA}E_{EST_{t-1}}}{E_{MEA}+E_{EST_{t-1}}} \Rightarrow [1-KG] E_{EST_{t-1}}\]

current.error = ( measured.error * previous.error ) / 
                ( measured.error + previous.error )

current.error = (1 - kalman.gain) * previous.error